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Properties

Label 2-440-440.59-c0-0-0
Degree $2$
Conductor $440$
Sign $0.794 - 0.606i$
Analytic cond. $0.219588$
Root an. cond. $0.468602$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.190 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (0.309 + 0.951i)11-s + (1.30 + 0.951i)13-s + (0.190 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)18-s + (−0.5 + 1.53i)19-s + (−0.809 − 0.587i)20-s + (0.309 − 0.951i)22-s + 0.618·23-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.190 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (0.309 + 0.951i)11-s + (1.30 + 0.951i)13-s + (0.190 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)18-s + (−0.5 + 1.53i)19-s + (−0.809 − 0.587i)20-s + (0.309 − 0.951i)22-s + 0.618·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.794 - 0.606i$
Analytic conductor: \(0.219588\)
Root analytic conductor: \(0.468602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :0),\ 0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5060996033\)
\(L(\frac12)\) \(\approx\) \(0.5060996033\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30253532847578394115441856340, −10.76516345709444848377205198304, −9.535743269777322079230704123905, −8.774568398587416298084908474003, −8.048381792993070636340859293257, −6.95696077081742688410481928523, −6.08895901824075764272385281885, −4.16650527135973297854551414472, −3.35739528430371123996751629381, −1.90926833830897970862286288612, 0.942359534289069936298232069661, 3.14725705661514812305844321937, 4.68677146207509353573017999424, 5.66630941715995649230358771886, 6.74761370673886706917019624374, 7.86416696857003306102926389911, 8.510290383492235168048121009058, 8.984916311161013413862630834388, 10.55172311454319624934468491494, 11.05480744611426148692345872033

Graph of the $Z$-function along the critical line